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Many problems in computational science and engineering can be described in terms of approximating a smooth function of $d$ variables, defined over an unknown domain of interest $\Omega\subset \mathbb{R}^d$, from sample data. Here both the curse of dimensionality ($d\gg 1$) and the lack of domain knowledge with $\Omega$ potentially irregular and/or disconnected are confounding factors for sampling-based methods. Na\"{i}ve approaches often lead to wasted samples and inefficient approximation schemes. For example, uniform sampling can result in upwards of 20\% wasted samples in some problems. In surrogate model construction in computational uncertainty quantification (UQ), the high cost of computing samples needs a more efficient sampling procedure. In the last years, methods for computing such approximations from sample data have been studied in the case of irregular domains. The advantages of computing sampling measures depending on an approximation space $P$ of $\dim(P)=N$ have been shown. In particular, such methods confer advantages such as stability and well-conditioning, with $\mathcal{O}(N\log(N))$ as sample complexity. The recently-proposed adaptive sampling for general domains (ASGD) strategy is one method to construct these sampling measures. The main contribution of this paper is to improve ASGD by adaptively updating the sampling measures over unknown domains. We achieve this by first introducing a general domain adaptivity strategy (GDAS), which approximates the function and domain of interest from sample points. Second, we propose adaptive sampling for unknown domains (ASUD), which generates sampling measures over a domain that may not be known in advance. Then, we derive least squares techniques for polynomial approximation on unknown domains. Numerical results show that the ASUD approach can reduce the computational cost by as 50\% when compared with uniform sampling.

Our goal is to develop theory and algorithms for establishing fundamental limits on performance for a given task imposed by a robot's sensors. In order to achieve this, we define a quantity that captures the amount of task-relevant information provided by a sensor. Using a novel version of the generalized Fano inequality from information theory, we demonstrate that this quantity provides an upper bound on the highest achievable expected reward for one-step decision making tasks. We then extend this bound to multi-step problems via a dynamic programming approach. We present algorithms for numerically computing the resulting bounds, and demonstrate our approach on three examples: (i) the lava problem from the literature on partially observable Markov decision processes, (ii) an example with continuous state and observation spaces corresponding to a robot catching a freely-falling object, and (iii) obstacle avoidance using a depth sensor with non-Gaussian noise. We demonstrate the ability of our approach to establish strong limits on achievable performance for these problems by comparing our upper bounds with achievable lower bounds (computed by synthesizing or learning concrete control policies).

Reinforcement Learning (RL) has the promise of providing data-driven support for decision-making in a wide range of problems in healthcare, education, business, and other domains. Classical RL methods focus on the mean of the total return and, thus, may provide misleading results in the setting of the heterogeneous populations that commonly underlie large-scale datasets. We introduce the K-Heterogeneous Markov Decision Process (K-Hetero MDP) to address sequential decision problems with population heterogeneity. We propose the Auto-Clustered Policy Evaluation (ACPE) for estimating the value of a given policy, and the Auto-Clustered Policy Iteration (ACPI) for estimating the optimal policy in a given policy class. Our auto-clustered algorithms can automatically detect and identify homogeneous sub-populations, while estimating the Q function and the optimal policy for each sub-population. We establish convergence rates and construct confidence intervals for the estimators obtained by the ACPE and ACPI. We present simulations to support our theoretical findings, and we conduct an empirical study on the standard MIMIC-III dataset. The latter analysis shows evidence of value heterogeneity and confirms the advantages of our new method.

It is common practice to use Laplace approximations to compute marginal likelihoods in Bayesian versions of generalised linear models (GLM). Marginal likelihoods combined with model priors are then used in different search algorithms to compute the posterior marginal probabilities of models and individual covariates. This allows performing Bayesian model selection and model averaging. For large sample sizes, even the Laplace approximation becomes computationally challenging because the optimisation routine involved needs to evaluate the likelihood on the full set of data in multiple iterations. As a consequence, the algorithm is not scalable for large datasets. To address this problem, we suggest using a version of a popular batch stochastic gradient descent (BSGD) algorithm for estimating the marginal likelihood of a GLM by subsampling from the data. We further combine the algorithm with Markov chain Monte Carlo (MCMC) based methods for Bayesian model selection and provide some theoretical results on the convergence of the estimates. Finally, we report results from experiments illustrating the performance of the proposed algorithm.

Mixtures of ranking models are standard tools for ranking problems. However, even the fundamental question of parameter identifiability is not fully understood: the identifiability of a mixture model with two Bradley-Terry-Luce (BTL) components has remained open. In this work, we show that popular mixtures of ranking models with two components (Plackett-Luce, multinomial logistic model with slates of size 3, or BTL) are generically identifiable, i.e., the ground-truth parameters can be identified except when they are from a pathological subset of measure zero. We provide a framework for verifying the number of solutions in a general family of polynomial systems using algebraic geometry, and apply it to these mixtures of ranking models. The framework can be applied more broadly to other learning models and may be of independent interest.

In online learning problems, exploiting low variance plays an important role in obtaining tight performance guarantees yet is challenging because variances are often not known a priori. Recently, considerable progress has been made by Zhang et al. (2021) where they obtain a variance-adaptive regret bound for linear bandits without knowledge of the variances and a horizon-free regret bound for linear mixture Markov decision processes (MDPs). In this paper, we present novel analyses that improve their regret bounds significantly. For linear bandits, we achieve $\tilde O(d^{1.5}\sqrt{\sum_{k}^K \sigma_k^2} + d^2)$ where $d$ is the dimension of the features, $K$ is the time horizon, and $\sigma_k^2$ is the noise variance at time step $k$, and $\tilde O$ ignores polylogarithmic dependence, which is a factor of $d^3$ improvement. For linear mixture MDPs with the assumption of maximum cumulative reward in an episode being in $[0,1]$, we achieve a horizon-free regret bound of $\tilde O(d \sqrt{K} + d^2)$ where $d$ is the number of base models and $K$ is the number of episodes. This is a factor of $d^{3.5}$ improvement in the leading term and $d^7$ in the lower order term. Our analysis critically relies on a novel elliptical potential `count' lemma. This lemma allows a novel regret analysis in conjunction with the peeling trick, which is of independent interest.

Gaussian process hyperparameter optimization requires linear solves with, and log-determinants of, large kernel matrices. Iterative numerical techniques are becoming popular to scale to larger datasets, relying on the conjugate gradient method (CG) for the linear solves and stochastic trace estimation for the log-determinant. This work introduces new algorithmic and theoretical insights for preconditioning these computations. While preconditioning is well understood in the context of CG, we demonstrate that it can also accelerate convergence and reduce variance of the estimates for the log-determinant and its derivative. We prove general probabilistic error bounds for the preconditioned computation of the log-determinant, log-marginal likelihood and its derivatives. Additionally, we derive specific rates for a range of kernel-preconditioner combinations, showing that up to exponential convergence can be achieved. Our theoretical results enable provably efficient optimization of kernel hyperparameters, which we validate empirically on large-scale benchmark problems. There our approach accelerates training by up to an order of magnitude.

In today's technology environment, information is abundant, dynamic, and heterogeneous in nature. Automated filtering and prioritization of information is based on the distinction between whether the information adds substantial value toward one's goal or not. Contextual multi-armed bandit has been widely used for learning to filter contents and prioritize according to user interest or relevance. Learn-to-Rank technique optimizes the relevance ranking on items, allowing the contents to be selected accordingly. We propose a novel approach to top-K rankings under the contextual multi-armed bandit framework. We model the stochastic reward function with a neural network to allow non-linear approximation to learn the relationship between rewards and contexts. We demonstrate the approach and evaluate the the performance of learning from the experiments using real world data sets in simulated scenarios. Empirical results show that this approach performs well under the complexity of a reward structure and high dimensional contextual features.

Contextual multi-armed bandit (MAB) achieves cutting-edge performance on a variety of problems. When it comes to real-world scenarios such as recommendation system and online advertising, however, it is essential to consider the resource consumption of exploration. In practice, there is typically non-zero cost associated with executing a recommendation (arm) in the environment, and hence, the policy should be learned with a fixed exploration cost constraint. It is challenging to learn a global optimal policy directly, since it is a NP-hard problem and significantly complicates the exploration and exploitation trade-off of bandit algorithms. Existing approaches focus on solving the problems by adopting the greedy policy which estimates the expected rewards and costs and uses a greedy selection based on each arm's expected reward/cost ratio using historical observation until the exploration resource is exhausted. However, existing methods are hard to extend to infinite time horizon, since the learning process will be terminated when there is no more resource. In this paper, we propose a hierarchical adaptive contextual bandit method (HATCH) to conduct the policy learning of contextual bandits with a budget constraint. HATCH adopts an adaptive method to allocate the exploration resource based on the remaining resource/time and the estimation of reward distribution among different user contexts. In addition, we utilize full of contextual feature information to find the best personalized recommendation. Finally, in order to prove the theoretical guarantee, we present a regret bound analysis and prove that HATCH achieves a regret bound as low as $O(\sqrt{T})$. The experimental results demonstrate the effectiveness and efficiency of the proposed method on both synthetic data sets and the real-world applications.

This work considers the problem of provably optimal reinforcement learning for episodic finite horizon MDPs, i.e. how an agent learns to maximize his/her long term reward in an uncertain environment. The main contribution is in providing a novel algorithm --- Variance-reduced Upper Confidence Q-learning (vUCQ) --- which enjoys a regret bound of $\widetilde{O}(\sqrt{HSAT} + H^5SA)$, where the $T$ is the number of time steps the agent acts in the MDP, $S$ is the number of states, $A$ is the number of actions, and $H$ is the (episodic) horizon time. This is the first regret bound that is both sub-linear in the model size and asymptotically optimal. The algorithm is sub-linear in that the time to achieve $\epsilon$-average regret for any constant $\epsilon$ is $O(SA)$, which is a number of samples that is far less than that required to learn any non-trivial estimate of the transition model (the transition model is specified by $O(S^2A)$ parameters). The importance of sub-linear algorithms is largely the motivation for algorithms such as $Q$-learning and other "model free" approaches. vUCQ algorithm also enjoys minimax optimal regret in the long run, matching the $\Omega(\sqrt{HSAT})$ lower bound. Variance-reduced Upper Confidence Q-learning (vUCQ) is a successive refinement method in which the algorithm reduces the variance in $Q$-value estimates and couples this estimation scheme with an upper confidence based algorithm. Technically, the coupling of both of these techniques is what leads to the algorithm enjoying both the sub-linear regret property and the asymptotically optimal regret.